232 Proceedings of Royal Society of Edinburgh. [sess. 
author’s machine was 1 inch, and the distance between them and 
the fixed pulleys was about 8 feet. After the mathematical theory 
had been worked out, these dimensions were reduced to their 
present values (J inch and 30 inches respectively), without 
impairing in any way the mechanical working. How far it may be 
possible to go in this direction it remains for further experiment 
to decide, but it certainly does not seem as if the limit had yet 
been reached. 
Another line of improvement is suggested by the following 
considerations. 
In the above discussion the two fixed pulleys in fig. 6 were 
considered to be at a distance apart equal to their effective 
diameters, but it is by no means certain that this distance is the 
best for the purpose, and another might be found, by means of 
a mathematical discussion or otherwise, for which the deviation 
of the pen would be much reduced. 
The author is continuing to investigate this subject, and hopes 
before long to be in a position to make a further report. He 
desires to express his best thanks to Professor MacGregor for 
the opportunity afforded to him of carrying on the work in the 
Physical Laboratory of the University of Edinburgh, and also for 
the loan of apparatus, etc. ; and to the Trustees of the Moray 
Fund for a grant obtained through Professor Chrystal for the 
construction of the present instrument. 
§ 12. Summary of Paper. 
The apparatus described in this paper is designed for the 
purpose of drawing the curve which is the summation of a number 
of simple harmonics, and it is so constructed that both the 
amplitude and the period of any of its constituent harmonics can 
be set to all values, commensurable or incommensurable, throughout 
their range. Further, the amplitude and the period of any of the 
constituent harmonics can be altered at will while the machine is 
in motion. 
The curves drawn by the pen of the instrument, although in 
theory not truly harmonic, are yet more than sufficiently so in 
practice, for it is shown that the greatest departure anywhere 
from the truth cannot be more than x -|q of an inch ; besides 
